The inner (dot) product of two real vectors \(u\) and \(v\) is \(u \cdot v = \sum_i u_i v_i = u^T v\); it equals zero precisely when \(u\) and \(v\) are orthogonal. From it comes the Euclidean norm \(\|x\| = \sqrt{x \cdot x} = \sqrt{\sum_i x_i^2}\). The Cauchy–Schwarz inequality \(|u \cdot v| \leq \|u\| \|v\|\) follows naturally, with equality if and only if \(u\) and \(v\) are linearly dependent. Hölder's inequality extends this to conjugate exponents: for \(1/p + 1/q = 1\), one has \(\sum_i |u_i v_i| \leq \|u\|_p \, \|v\|_q\).
A vector norm in general is a function on a space that is nonnegative, vanishes only at the zero vector, scales by scalars \(\|c x\| = |c| \|x\|\), and satisfies the triangle inequality \(\|u + v\| \leq \|u\| + \|v\|\). For matrices, the Frobenius norm is \(\|A\|_F = \sqrt{\sum_{i,j} A_{ij}^2} = \sqrt{\sum \sigma_i^2}\) where \(\sigma_i\) are the singular values, while the L2 or spectral norm \(\|A\|_2\) is the largest singular value — the maximum factor by which \(A\) stretches any unit vector.
The condition number \(\kappa(A) = \|A\| \cdot \|A^{-1}\|\), equal in the 2-norm to the ratio of the largest to smallest singular value, captures how badly a linear system amplifies perturbations. A large condition number means that small changes in \(b\) can produce large changes in \(x = A^{-1} b\), making numerical solutions unreliable. The spectral radius \(\rho(A)\) is the largest absolute eigenvalue, and Gershgorin's circle theorem locates every eigenvalue in at least one disc centered at a diagonal entry \(a_{ii}\) with radius equal to the sum of absolute off-diagonal entries in row \(i\).
When given a basis that is not orthogonal, one can build one that is. The Gram–Schmidt process orthogonalizes a basis by successively subtracting projections onto previously constructed vectors and normalizing; it produces a QR decomposition \(A = QR\) where \(Q\) has orthonormal columns and \(R\) is upper triangular. Vectors that are mutually orthogonal and each of unit length are called orthonormal, and such collections form natural coordinate systems in which calculations mimic those in the standard basis.